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A093682
Array T(m,n) by antidiagonals: nonarithmetic-3-progression sequences with simple closed forms.
16
1, 2, 1, 4, 3, 1, 5, 4, 4, 1, 10, 6, 5, 7, 1, 11, 10, 8, 8, 10, 1, 13, 12, 10, 10, 11, 19, 1, 14, 13, 13, 11, 13, 20, 28, 1, 28, 15, 14, 16, 14, 22, 29, 55, 1, 29, 28, 17, 17, 20, 23, 31, 56, 82, 1, 31, 30, 28, 20, 22, 28, 32, 58, 83, 163, 1, 32, 31, 31, 28, 23, 29, 37, 59, 85
OFFSET
0,2
COMMENTS
The nonarithmetic-3-progression sequences starting with a(1)=1, a(2)=1+3^m or 1+2*3^m, m >= 0, seem to have especially simple 'closed' forms. None of these formulas have been proved, however.
T(m,1)=1, T(m,2) = 1 + (1 + [m even])*3^floor(m/2) = 1 + A038754(m), m >= 0, n > 0; T(m,n) is least k such that no three terms of T(m,1), T(m,2), ..., T(m,n-1), k form an arithmetic progression.
FORMULA
T(m, n) = (Sum_{k=1..n-1} (3^A007814(k) + 1)/2) + f(n), with f(n) a P-periodic function, where P <= 2^floor((m+3)/2) (conjectured and checked up to m=13, n=1000).
The formula implies that T(m, n) = b(n-1) where b(2n) = 3b(n) + p(n), b(2n+1) = 3b(n) + q(n), with p, q sequences generated by rational o.g.f.s.
EXAMPLE
Array begins:
1, 2, 4, 5, 10, 11, 13, ...
1, 3, 4, 6, 10, 12, 13, ...
1, 4, 5, 8, 10, 13, 14, ...
1, 7, 8, 10, 11, 16, 17, ...
1, 10, 11, 13, 14, 20, 22, ...
...
CROSSREFS
Column 2 is 1+A038754. Cf. A092482, A033158.
Sequence in context: A389359 A378037 A382252 * A344767 A187883 A134543
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Apr 09 2004
STATUS
approved